Discussion Paper No.
590
ENDOGENOUS HUMAN CAPITAL
ACCUMULATION, COMPARATIVE ADVANTAGE
AND DIRECT
VS. INDIRECT REDISTRIBUTION
Hisahiro Naito
July 2003
The Institute of Social and Economic Research Osaka University
Endogenous Human Capital Accumulation, Comparative
Advantage and Direct vs. Indirect Redistribution
Hisahiro Naito ∗†
Institute of Social and Economic Research Osaka University
and
Department of Economics University of California Irvine Previous version August 30, 2002
Current version June 7, 2003
Forthcoming from Journal of Public Economics
Abstract
Recently, several papers have re-examined the so-called production efficiency theorem and the Atkinson and Stiglitz theorem on commodity taxes in the optimal taxation literature. Naito (1999) showed that indirect redistribution through production distortion or consump-tion distorconsump-tion can Pareto-improve welfare and that the two theorems do not necessarily hold when different factors are imperfect substitutes and factor prices are endogenous. On the other hand, Saez (2001) argued that in the long run where human capital accumulation is endogenous, the two theorems are still valid. This paper develops reasonable alternative mod-els where individuals accumulate human capital based on their comparative advantage. The present paper shows that the production efficiency theorem is not necessarily valid and that indirect redistribution from the able to the less able such as tariffs and production subsidies can increase efficiency even when skill accumulation is endogenous.
Keywords: Human capital accumulation, non-linear income taxation, and comparative advantage JEL Number: H21, H23
∗
I appreciate the participants of the Public Economics Research Group around Osaka and Daiji Kawaguchi for his helpful information on the relationship between earnings and ability. I am very grateful for the comments from two anonymous referees and professor Robin Boadway, the editor of the Journal. Their comments have been very helpful for improving this paper. Of course, the author is responsible for all remaining errors.
†
Address: Institute of Social and Economic Research, Osaka University, Mihogaoka 6-1, Ibaraki City, Osaka, Japan, postal code 567-0047
phone:81-6-6879-8581; fax: 81-6-6878-2766.
1
Introduction
Whether efficient income redistribution should be done through income taxation alone or should
be complemented with other measures such as production distortion or consumption distortion
is one of the key issues whenever optimal public policies are discussed. With this regard, the
production efficiency theorem (Diamond and Mirrlees, 1971), which states that production
dis-tortion is not optimal and the Atkinson and Stiglitz theorem on optimal commodity taxation
(Atkinson and Stiglitz, 1976, 1980), which shows that commodity taxation is not necessary in
the presence of an optimal income tax system, are the most important results in public finance
literature.
However, in public finance literature researchers started examining those results. For example,
Cremer, Pestieau and Rochet (2001) showed that the Atkinson and Stiglitz theorem does not hold
when individuals are different in ability and endowment. Saez (2003) showed that the Atkinson
and Stiglitz theorem does not hold when tastes are heterogenous. Naito (1999) showed that in
a model similar to the model of Stiglitz (1982), if multiple goods are produced and factor prices
are endogenous, the Atkinson and Stiglitz theorem does not necessarily hold and the production
efficiency result does not either.
On the other hand, many of the previous studies on optimal income taxation have received
criticism that they did not focus on long term decisions such as human capital accumulation but
focused on the short term choices such as labor supply. As a result, it is sometimes argued that
the result obtained in the short run model might not hold in the long run.
In particular, Saez (2003) made skill accumulation endogenous in the model of optimal
tax-ation and analyzed several issues of public policy. He showed that Naito’s results are not valid
and that the production efficiency theorem and the Atkinson and Stiglitz theorem on
commod-ity taxation are valid when human capital accumulation is endogenous. Since accumulation of
human capital has a strong effect on the economy in the long run and since the implications of
contribution of Saez’s paper is substantial.
Despite such contributions, however, we believe that a further investigation would be needed.
In many previous analyses involving asymmetric information not only in public finance literature
but also in other literature, conclusions were not often robust in the sense that they critically
depended on the structure of information and the timing of information revelation. Thus, it
is worthwhile investigating the robustness of the result of Saez (2003) with another reasonable
set of assumptions. In particular, in this paper we will show that if higher ability persons have
comparative advantage in the sense that the relative return from accumulating skilled human
capital to unskilled human capital is higher than that of lower ability persons, the production
efficiency theorem does not hold. 1
To explain the intuition of the present paper, it would be useful to look at the differences
between the assumptions in Saez (2003) and those in Naito (1999). In Naito (1999), there are
two factors of production that are imperfect substitutes. In addition, from the beginning, each
individual is attached to a particular labor market (the skilled labor market or the unskilled
labor market) but the government cannot observe whether each individual is attached to the
skilled or unskilled labor markets. The main idea in Naito (1999) is that when the government
cannot observe an individual’s type, the government can affect different individuals differently by
using the response of the factor markets (Stolper and Samuelson theorem, Stolper and Samuelson
(1941)). Since the income tax policy cannot discriminate the different types of agents attached
to different labor markets but a commodity tax and a tariff can, using a commodity tax (in a
case of a closed economy), or a tariff (in a case of an open economy) with the response of factor
markets can increase the efficiency.
In Saez (2003), each job requires pre-determined skill levels. As a result, the income level
1In this paper, we only analyze the case of a small open economy due to the limitation of the space. As a result,
we only prove that the production efficiency theorem does not hold. In a small open economy, a commodity tax cannot affect the producer prices and hence, factor prices. Therefore, the Atkinson and Stiglitz theorem holds in a small open economy. On the other hand, the result is changed in a closed economy. In the previous version of the paper, we proved that the Atkinson and Stiglitz theorem does not hold in a closed economy. See the section 5 in the present paper for more discussion. Also, for proof, please see our previous version of the present paper (Naito, 2002).
represents the amount of skill the individuals acquired. People have a heterogenous ability to
acquire skill. However, since such heterogeneity of ability is incorporated in the utility function
as a difference of disutility to acquire skill, there is not any room for such heterogenous ability to
interact with the market reaction. Thus, the heterogeneity of abilities is intrinsically independent
of the external environment of the economy. When the heterogeneity of abilities is independent
of the response of the factor market, it is essentially equivalent to assuming that the dimension
of factors used for production is one. In such a case, changes of factor prices due to government
policy cannot increase the economic efficiency.
The key idea of the present paper is that in the presence of comparative advantage in
ac-cumulating different types of human capital, individuals with different abilities will be affected
differently by the responses of factor markets even when skill accumulation is endogenous. In
such a case, a policy that introduces inefficient production but affects the factor prices differently
for different factors can indirectly redistribute from the less able to the able. Although such a
policy cause a distortion, it has only the second order effect, but such an indirect redistribution
has the first order effect on welfare. Thus, it can increase the social welfare.
For illustration, consider a situation where there are two types of human capital: skilled human
capital and unskilled human capital and where those who have higher ability have comparative
advantagein accumulating skilled human capital. Comparative advantagein accumulating skilled
human capital for the able means that the relative benefit from accumulating skilled human
capital to unskilled human capital for individuals with high ability is higher than for the less
able. We could think that training, knowledge and experience in white collar jobs are skilled
human capital and those in blue collar jobs are unskilled human capital. In such a situation, a
decrease of the return from skilled human capital and an increase of the return from unskilled
human capital will hurt the able relatively more and give relatively more benefit to the less able.
The intuition of this paper is that when individual ability is not observable to the social planner
individuals, then a policy that will change the returns from skilled and unskilled human capital
differently might be useful for an efficiency reason.
The crucial assumption in the present paper is the presence of comparative advantage in
human capital accumulation. Whether such an assumption is reasonable or not is an interesting
empirical question. Earlier literature of the human capital theory assumed that earning could be
explained completely once it is conditioned by human capital level. Earlier empirical evidences
showed that there is a strong correlation between earnings and the level of human capital and
indicated that ability does not matter for explaining earnings once they are conditioned by the
human capital levels. On the other hand, recent literature of labor economics and self-selection
emphasizes that ability can also increase earning and play a systematic role for explaining earnings
even after it is conditioned by the human capital level. This literature points out that even in
an extreme case when human capital does not increase the productivity at all, if ability can
increase the productivity and if higher ability agents tend to acquire more skills, there will be a
correlation between human capital level and earnings. In the standard signaling literature, it is
commonly assumed that a higher ability person would get more benefit from acquiring skill. In
addition, recently, Dinardo and Tobias (2001) and Tobias (2003) examined whether the returns
from schooling are higher for high ability individuals than for low ability individuals by using a
non-parametric method. They found that the returns are higher for high ability individuals than
for low ability individuals. This suggests that assuming the presence of comparative advantage
is not unrealistic as an approximation of the reality.
At this point, one might wonder about the difference between Naito (1999) and the present
paper. In the case of Naito (1999), each type of worker is attached to a different labor market.
As a result, skilled workers can supply only skilled labor and unskilled workers can supply only
unskilled labor. However, in the present paper, both high ability persons and low ability persons
have options to accumulate both types of human capital or either type of human capital. Thus,
return from skilled human capital always increases efficiency is not obvious.
The organization of this paper is as follows. In section 2, we present the model in a small
open economy and analyze the production efficiency theorem by Diamond and Mirrlees (1971)
when two factors are imperfect substitutes. In section 3, we analyze the same issue when two
factors are perfect substitutes. In section 4, we shall give the implications and in section 5 we
will give a brief conclusion.
2
The model
The economy is small and open and there are two output goods: good 1 and good 2. Good 1 is
skilled human capital intensive good and good 2 is unskilled human capital intensive good. We
assume that there are two types of human capital in this economy: skilled human capital and
unskilled human capital. In this economy, there is a continuum of agents and all agents have
identical, additive separable utility functions with respect to consumption, skilled human capital
investment and unskilled human capital investment. We index all individuals’ ability by iwhere
i takes any value from one to two. We assume that the utility function of the typei agent has
the following form:
u(c1i, c2i)−fs(hsi)−fu(hui)
where u(c1i, c2i) is strictly increasing with each argument and strictly concave and fs(hsi) and
f(hui) are strictly increasing and strictly convex. c1i and c2i are the consumption of good 1 and
good 2 by agent i. We assume that the labor supply is fixed and it is normalized to one. hsi
and hui are the levels of skilled and unskilled human capital of individual i. hsi and hui can be
interpreted as the knowledge levels, years of education, experience and training for each type of
skill. In addition, to illustrate our point, we assume that fs(hsi) and f(hui) have the following
functional forms:2
fs(hsi) = (hsi)γs andfs(hui) = (hui)γu 2
whereγs and γu measure the curvature of the disutility functions of skilled and unskilled human
capital accumulation respectively and they are strictly greater than one. Given the amount of
skilled human capital and unskilled human capital of individuali, we assume that the earning of
individual i is determined as follows:
earningi =gs(i)×ws×his+gu(i)×wu×hui (1)
where ws and wu are the returns from one efficient unit of skilled and unskilled human capital,
respectively. (1) means that when individual i accumulates hsi units of skilled human capital
and hui units of unskilled human capital, the efficient unit of skilled human capital and unskilled
human capital are gs(i)×hsi and gu(i)×hui and the total return from skilled human capital and
unskilled human capital are gs(i)×ws×hsi and gu(i)×wu ×hsi, respectively. Let gs(i)×ws
and gu(i)×wu be wis and wui. gs0(i)/gs(i) and g0u(i)/gu(i) measure the absolute advantage of an
agent with abilityi+over agentiin accumulating skilled human capital and unskilled human
capital, respectively. We assume that agents who have higher ability have absolute advantage
in accumulating both skilled human capital and unskilled human capital: g0s(i)/gs(i) > 0 and
g0u(i)/gu(i). 3 Also, as we discussed in the introduction, we assume that agents who have higher
ability have comparative advantage in accumulating skilled human capital than unskilled human
capital. Thus, we assume that
gs0(i) gs(i) > g 0 u(i) gu(i) γu γs (2)
The assumption (2) has a clear economic meaning. Consider a situation where the disutility
functions of accumulating skilled human capital and unskilled human capital have the same
degree of curvature (γs = γu). In this case, (2) means that an agent whose ability is higher
will have a larger rate of increase of ws
i, the return from accumulating skilled human capital,
than that of wiu, the return from accumulating unskilled human capital. When the curvature of
the disutility functions are different, (2) says that the condition of the comparative advantage
3
The assumption of the absolute advantage is not necessary. The assumption of the absolute advantage is a sufficient condition that guarantees that agents who have higheriwill receive higher utility. As long as agents with higher ability can receive higher utility the assumption of the absolute advantage is not necessary.
must be adjusted by the ratio of the curvatures of marginal disutility of skilled human capital
accumulation and unskilled human capital accumulation. 4
At this point, note that (1) is different from the assumptions in Saez in several ways. In
Saez, he assumed that heterogeneity of individuals is incorporated in the utility function, not
the earning equation. Thus, once earning is conditioned by the human capital level, individual
level heterogeneity of ability does not play any systematic role for explaining earnings. On the
other hand, in (1) even after conditioned by the level of human capital, heterogeneity of ability
plays a systematic role for explaining earnings and it induces higher earning for agents with
higher ability. In addition, the relative return from one efficient unit of skilled human capital to
unskilled human capital is higher for the agent who has higher ability than for the agent who has
lower ability. As we discussed in the introduction, this interaction term between heterogeneity of
ability and the return of human capital plays a crucial role in the present paper.
As for the objective of the government, we assume that the social planner will maximize the
following utilitarian social welfare function:
Z 2
1
{u(c1i, c2i)−fs(his)−fu(hui)}nidi . (3)
As for prices, we normalize the producer price and the consumer price of good 1 to one.
Let p2, q2 and p∗2 be the consumer price and the producer price and the international price of
good 2, respectively. As the purpose of this section is to examine whether introducing production
distortion can increase the social welfare or not, we consider imposing a tariff on good 2. Although
a tariff introduces not only a production distortion but also a consumption distortion, the first
order effect of consumption distortion on welfare can be ignored as we will demonstrate. Let σ
4
The reason that the terms ofgs0/gsandg0u/guneed to be adjusted by the curvature of the marginal disutility
is as follows. For illustration, consider a condition thatgs0/gs and g0u/gu must satisfy when agents with ability
i+and agents with abilityihave the same degree of comparative advantage under the assumption ofγs > γu.
The assumption ofγs > γu implies that the marginal disutility of skilled human capital changes faster than the
marginal disutility of unskilled human capital when the amount of skilled and unskilled human capital respectively changes at the same rate. Note that the marginal disutility per return of skilled and unskilled human capital must be equal at the margin. This implies that in order that agents with abilityi+and agents with abilityihave the same degree of comparative advantage,g0s/gs must be smaller thangu0/gu.
be a size of a tariff on good 2. Then, we will have
p2=q2 =p∗2+σ. (4)
As for the equations determining the returns from skilled and unskilled human capital, we
assume the standard two sector Heckscher-Ohlin model. In this economy, there are two sectors.
Sector 1 is the skilled human capital intensive sector and it produces good 1. Sector 2 is the
unskilled human capital intensive sector and it produces good 2. Each sector uses both skilled and
unskilled human capital. Consumers (workers) are perfectly mobile between two sectors. When
an agent who hashsi units of skilled human capital andhui units of unskilled human capital works
in sector k, it means that sector k uses gs(i)×hsi units of skilled human capital and gu(i)×hui
units of unskilled human capital. Each sector behaves as a price taker and maximizes its profit.
Let Fk(Hs
k, Hku) be the production function in sector k = 1,2 where Hks and Hku are the total
amount of skilled human capital and unskilled human capital used in sector k. We assume that
Fk(Hks, Hku) exhibits constant returns to scale and it is concave with respect to both arguments.
Let ck(ws, wu) be the cost function in sector k to produce one unit of output in sector k when
the returns of one efficient unit of skilled human capital and unskilled human capital arews and
wu, respectively. When both good 1 and good 2 are produced at the equilibrium, ws andwu are
determined
1 =c1(ws, wu) andq2=c2(ws, wu), (5)
From the Stolper -Samuelson theorem,∂ws/∂q <0 and ∂wu/∂q >0.
The output of both goods are determined from the following factor market equilibrium
con-ditions: ∂c1 ∂wsy 1+ ∂c2 ∂wsy 2 = Z 2 1 gs(i)×hsi ×nidi, and ∂c1 ∂wuy 1+ ∂c2 ∂wuy 2 = Z 2 1 gu(i)×hui ×nidi (6)
Although the output of both goods can be calculated from equation (6), it is more useful to
work on the production possibility frontier for analytical reasons. Let Hs and Hu be the total
production possibility frontier as Γ(Hs, Hu). Since the production functions are concave and the
factor intensity of the two sectors are different, the production possibility set is convex. Let the
producer price of good 1 and good 2 be 1 and q2. Then, the output of good 1 and good 2 are
determined as the solution of the following constrained maximization problem:
max y1+q2y2 s.t. (y1, y2)∈Γ(Hs, Hu) = 0
Thus, the output of good 1 and good 2 can be thought as a function of q2, Hs and Hu. Let
Y(q, Hu, Hu) be the output function of good 2. At the optimum, the slope of production
possi-bility set is equal to the relative producer price of good 2. Thus, we obtain Yq ≡∂Y /∂q2 > 0.
The Rybcyzynski theorem shows thatYHu ≡∂Y /∂Hu >0 and YHs ≡∂Y /∂Hs<0.
The purpose of the social planner is to maximize the utilitarian social welfare function. Given
the additive separable utility function and the utilitarian social welfare function, the social planner
wants to redistribute income from those who have higher ability to those who have lower ability.
On the other hand, since the social planner cannot observe individual ability but rather individual
earning, the social planner needs to design a non-linear income tax system T(R) to redistribute
income where T(R) is a tax liability function andR is pre-tax income.
Before designing an income tax system, it is useful to consider the problem of designing the
non-linear income tax system in two steps. The first step is to know how an individual i will
choose skilled human capital and unskilled human capital to generate pre-tax income, R. The
second step is to know, given an after-tax-income schedule ofX=R−T(R), how each individual
chooses pre-tax income.
The first stage of the problem can be solved considering the following programming problem:
minfs(hsi) +fu(hui) (7)
s.t. R=wsi ×his+wui ×hui
where wis=gs(i)×ws and wui =gu(i)×wu
disutility to generate the pre-tax income R for an agent whose net returns from skilled human
capital and unskilled capital are wsi and wiu, respectively. We denote the solution of the above
problem as hsi(wsi, wui, R) and hui(wsi, wui, R).For the analysis later, it is useful to calculate
com-pensated human capital supply. Consider the following dual problem of (7):
E(wsi, wui, V)≡max wishsi +wiuhui
st. fs(hsi) +fu(hui)≤V
Let the solution of the above problem be ehji(wsi, wui, V) where j = s, u. Then, from the dual
relationship, we will have
hji(wsi, wui, E(wsi, wui, V))≡eh j i(w
s
i, wiu, V) j=s,u
By taking derivative on both sides, we will have the Slutsky equation forhsi and hui:
∂hji ∂wsi + ∂hji ∂Rh s i = ∂ehji ∂wsi and ∂hji ∂wiu + ∂hji ∂Rh u i = ∂ehji ∂wiu j=s,u
Note that the indifference curve of fs(hsi) +fu(hui) is strictly concave. Therefore, ∂ehsi/∂wsi >0,
∂ehui/∂wu < 0, ∂ehui/∂wui > 0 and ∂ehui/∂wsi < 0. This relationship means that if an individual
maximizes his earnings holding the total disutility constant, an increase of the net return from
skilled human capital will increase the supply of skilled human capital and an increase of the
return of unskilled human capital will decrease the supply for skilled human capital. As for
the properties of Z, let the Lagrangian multiplier of the disutility minimization problem beαi.
Then, fs0(hsi) = αiwsi , fu0(hui) = αiwu,Zws
i ≡ ∂Z/∂w
s
i = −αihsi, Zwu ≡ ∂Z/∂wu =−αi huiand
ZR≡∂Z/∂R =αi.
LetX(R) be the after-tax income schedule that the government designed. Then, at the second
stage of the problem, given Z(ws
i, wiu, R) andX(R), each individualiwill maximize his utility:
max
{R} U(p2, X(R))−Z(w
s
i, wiu, R)
where U(p2, x) is the indirect utility function from the consumption of two goods when the
is to design a schedule ofX(R) to maximize the social welfare. On the other hand, the Revelation
Principle shows that without loss of generality we can focus on the incentive compatible revelation
mechanism. Thus, let (Rj, Xj) be the pre-tax income and after tax income when an agent
announces that his type isj. Then define v(i) and bv(j;i) as follows:
v(i) = max {j} U(p2, Xj)−Z(w s i, wui, Rj) b v(j;i) =U(p2, Xj)−Z(wsi, wiu, Rj)
v(i) is the maximized utility given the schedule of (Rj, Xj) andbv(j;i) is the indirect utility when
agentiannounces that he is typej. The incentive compatibility condition implies that the type
iagent has an incentive to announce that he is type i:
i= arg max
{j} bv(j;i)
Assuming the differentiability of (Xj, Rj), the first order condition of the incentive compatibility
condition is ∂bv(j, i) ∂j j=i = ∂U ∂x ∂x ∂j − ∂Z ∂R ∂R ∂j = 0
On the other hand, by using the above first order condition, we havedv/di=−Zws i ×(dw
s i/di).
Sinceαi is the Lagrangian multiplier of the required income constraint in the disutility
minimiza-tion problem (7), from the FOC of the minimizaminimiza-tion problem we obtain
dv di =αiRi{ gs0(i) gs(i) θsi+ gu0(i) gu(i) θui} where θji = wjihji Ri (8)
Because of the assumption from the absolute advantage, dv/di > 0. (8) has a clear economic
meaning. It means that the slope of the value function v(i) is proportional to the weighted
average of the absolute advantage of skilled human capital accumulation and unskilled human
capital accumulation. For analytical reasons, it is useful to eliminate αi in the above equation.
Using the first order condition for hsi and hui, we can rewrite (8) as follows:
dv di = g0s(i) gs(i) fs0(hsi)hsi +g 0 u(i) gu(i) f0(hui)hui. (9)
Given (9), as Mirrlees (1971) pointed out, it is more useful to assume that the social planner
controls vi and Ri.5 Then,xi is defined by the following relationship:
v(i) =U(p2, Xi)−Z(wsi, wui, Ri). (10)
Let x(R, v, p2, wsi, wui) be the solution that solves (10) about X. Obviously, ∂x/∂v = (Ux)−1 ,
∂x/∂Ri = ZR/Ux and ∂x/∂p = −(Up)/(Ux), ∂x/∂p = −(Up)/(Ux) ,∂x/∂wsi = Zws
i/(Ux) and
∂x/∂wu =Zwu/(Ux).
Finally, the government budget constraint implies that
Z 2 1 ni{Ri−xi}di+σ{ Z 2 1 c2inidi−Y(p∗2+σ, Hs, Hu)}= 0
The problem of the social planner is to solve the following constrained optimization program:
W(σ) = max {Ri,vi} Z 2 1 v(i)nidi. st. dv di = gs0(i) gs(i) fs0(hsi)hsi +g 0 u(i) gu(i) f0(hui)hui Z 2 1 ni{Ri−xi(Vi}di+σ{ Z 2 1 c2inidi−Y(p∗2+σ, Hs, Hu)}= 0 σ is given.
In the above programming problem,W(σ) is the maximized social welfare for givenσ. Also note
thaths
i and hui are functions of (Ri, wis, wiu) and thatwsi and wui are the functions ofσ.
Our interest is to know whether a change of σ from 0 will increase the social welfare or not.
Analytically, by calculatingdW/dσ, and evaluating at σ = 0, we can check whether introducing
a distortion in production side (and consumption side too) can increase the social welfare. Letµi
andλbe the Lagrangian multiplier of the incentive compatibility constraint and the government
5
One might think that the local incentive compatibility constraints are not sufficient for the global incentive compatibility constraints. On the other hand, the literature of the mechanism design shows that a single crossing property (SCP) and the monotonicity constraints are sufficient conditions for local incentive compatibility con-straints to satisfy the global incentive compatibility concon-straints. (Fudenberg and Tirole, 1992). In this paper, we assume that the monotonicity constraints are always satisfied. This assumption is equivalent to assuming that there is no bunching. Many of the previous papers assumed that there is no bunching at the optimum. (Konishi 1995, Naito 1998). As for SCP, we can check it by examining ∂R∂i∂2Z >0. This is true as long as ∂hsi
budget constraint. By using the envelope theorem, we obtain dW dσ σ=0 =− Z 2 1 µi d[f0(hsi)hsi(gs0/gs)] dhs i {dh s i dws i dwis dσ + dhsi dwu dwui dσ }di − Z 2 1 µi d[f0(hu i)hui(gu0/gu)] dhui { dhu i dwiu dwu i dσ + dhu i dwiu dwu i dσ }di +λ{ Z 2 1 c2inidi−Y(p∗2+σ, Hs, Hu)}+λ Z 2 1 (−∂xi ∂p − ∂xi ∂ws i ∂wsi ∂σ − ∂xi ∂wu ∂wu ∂σ )nidi
After several calculations, we can obtain the following equation (See Appendix):
dW dσ σ=0 =− Z 2 1 µi n γs(g 0 s/gs)−γu(g 0 u/gu) o fs0(hsi)[∂eh s i ∂ws i ∂wsi ∂σ + ∂ehsi ∂wu i ∂wiu ∂σ ]di >0 (11)
Because of the property of the compensated supply function ofhs
i,∂ehsi/∂wsi >0 and∂ehsi/∂wui <
0. From the Stolper-Samuelson theorem,∂wsi/∂σ <0 and∂wui/∂σ >0. From the assumption on
comparative advantage, γs(g
0
s/gs)−γu(g
0
u/gu)>0. As for the sign of the Lagrangian multiplier
of the incentive compatibility constraint, the standard argument shows thatµi ≥0 for alli.(See
Appendix). Thus, we obtain dW/dσ >0.
Proposition 1 Suppose that at the zero distortion on production and the consumption in a small
open economy the social planner sets the income tax structure to maximize the social welfare
function in an endogenous skill accumulation model . Then an introduction of a tariff (export
subsidy) on an unskilled-labor-intensive good will increase the social welfare.
The above equation (11) has several implications. For an illustration, consider a situation
where the disutility functions of skilled and unskilled human capital accumulation have the same
degree of curvature, i.e. γs = γu ≡γ. Then, (11) shows that if (g
0
s/gs) = (g
0
u/gu), dW/dσ = 0.
In other words, if there is no comparative advantage and if higher ability individuals are as good
at accumulating skilled and unskilled human capital as lower ability individuals, then there is
no welfare gain from changing the returns of skilled and unskilled human capital. Second, note
that (∂ehsi/∂wis)(∂wsi/∂σ) and (∂ehsi/∂wui)(∂wiu/∂σ) measure how changes of returns from each
γ ×fs0(hsi) = fs00(hsi)hsi +fs0(hsi) and that fs00(hsi)hsi +fs0(hsi) is related with a change of ˙v. In
addition, note thatµimeasures how the social welfare increases when the incentive compatibility
is relaxed. This implies that the term after the integration measures how a compensated change
of the returns from skilled and unskilled human capital changes the slope of ˙v and increases the
social welfare. Also the calculation needed for obtaining the equation shows that the effect of
consumption distortion on welfare is zero, because as long as σ is small, such a distortion is of
the second order.6
The intuition of the above proposition is as follows. In a situation where higher ability
individuals have comparative advantage in accumulating skilled human capital and lower ability
individuals have comparative advantage in accumulating unskilled human capital, a decrease of
the return from skilled human capital and an increase of the return from unskilled human capital
will hurt higher ability individuals and benefit lower ability individuals. If the social planner is
interested in redistributing income from high ability individuals to low ability individuals, such
changes of the returns from skilled and unskilled capital can indirectly redistribute income. On
the other hand, starting from zero distortion, the deadweight loss of the production distortion
is of the second-order but the welfare gain of relaxing the incentive problem has the first-order
effect. As a result, introducing the production distortion increases the social welfare.
3
Extension: A case of Perfect substitute
In the previous section, we have assumed that two types of human capital are imperfect substitutes
in order to assume differentiability of the human capital accumulation functions. As a result,
people always accumulate both types of human capital. In reality, however, people sometimes
accumulate only one type of human capital and, as a result, the choice of human capital becomes
discrete. The purpose of this section is to analyze the welfare effect of direct versus indirect
redistribution when human capital accumulation is endogenous and different types of human
6
This can be easily checked fromλR2
1 c2inidi=λ R2
1(−
∂xi
capital are perfect substitutes. 7
In this section, because the assumption that two types of human capital are perfect substitutes,
we assume the following utility function for agenti:
u(c1i, c2i)−ashsi −auhui
whereu(c1i, c2i) is strictly increasing with each argument and strictly concave.
As in the previous section, we assume that following comparative advantage condition holds:
g0s(i) gs(i) > g 0 u(i) gu(i) (12)
The economic meaning of the above equation is the same as before. When two types of skill
ac-cumulation are perfect substitutes in the disutility function, the agent always solves the following
constrained disutility minimization problem:
Z(wsi, wui, R)≡minashsi +auhui
stR=wishsi +wuihui
wherewsi =gs(i)×ws andwui =gu(i)×wu
In the above problem, for an agent with abilityi, ifas/au < wsi/wiuhe will accumulate only skilled
human capital and if as/au > wsi/wiu, he will accumulate only unskilled human capital. Note
that because of the assumption of comparative advantage (12),wis/wiu is an increasing function
of i. Let i∗ be ithat satisfies (ws×gs(i))/(wu×gu(i)) =as/au. Then, agents whose ability is
greater thani∗ accumulate only skilled human capital and agents whose abilityi is less than i∗
accumulate only unskilled human capital. We assume that such i∗ is located within 1 and 2.8
7Besides the reason mentioned in the previous section, conducting a welfare analysis when individual behavior
includes a discrete choice is useful from a theoretical standpoint as well. In many important economic situations such as the choice of location to live, the choice of technology by firms and labor market participation, decisions made by consumers or firms include discrete choices. Until very recently, a welfare analysis that includes discrete choices was rare. As far as the author knows, only Boadway and Cuff (2001) started to investigate this issue very recently. They analyzed an optimal taxation problem when some individuals are bunched at the bottom. Another purpose of this section is to contribute to such a literature as well.
8This assumption is not so restrictive as the following reason. For example, ifi∗
Given suchi∗ , Z(wis, wiu, R) is Z(wsi, wui, R) =as(R ws i ) for i∗ ≤i≤2 Z(wsi, wui, R) =au( R wiu) for 1≤ i < i ∗.
LetX(R) be an after-tax income schedule that the government designs. Then, each agent chooses
his best R to maximize U(p2, X(R))−Z(wis, wui, R). Once R is chosen, an agent chooses his
optimal skill type and accumulates human capital to generate pre-tax incomeR. Letev(i) be the
maximized value given the schedule X(R):
e
v(i)≡max
R U(p2, X(R))−Z(w s
i, wiu, R).
For the analysis of the optimal schedule of X(R), we assume that the schedule of X(R) is a
continuous function. Although it is possible that the optimal schedule ofX(R) is not continuous,
the tax schedules of almost of all developed countries are continuous. When X(R) is a
contin-uous function, it is straightforward to show that ev(i) is continuous with respect to i from the
theory of the maximum (Berg 1963). In addition, there is an interesting property onev(i) in the
neighborhood of i∗ that turns out to be crucial for our result. The following lemma shows that
property of ev(i).
Lemma 1 When iincreases, the graph of ev(i) has a counter-clockwise kink at i∗.
Proof. Let evs(i) be the maximized utility of an agent with abilityigiven the tax schedule when
he can accumulate only skilled human capital. Also, let evu(i) be the maximized utility of an
agent with abilityiwhen he can accumulate only unskilled human capital. By the definition, the
graph of ev(i) is the upper envelope of evs(i) and evu(i) and i
∗ is at the intersection between
e vs(i)
and evu(i). This implies that there is a counter-clockwise kink ati∗ (See also Figure 1).
will accumulate only unskilled human capital. However, the production needs both skilled and unskilled human capital. As a result, the return from skilled human capital will start to increase and the return from unskilled human capital will start to decrease. This implies thati∗will start to decrease. This process will continue until some agents start to accumulate skilled human capital.
Now consider the problem of designing a nonlinear income tax system. As in the previous
section, we define v(i) as follows:
v(i) = max
j U(p2, Xj)−Z(w s
i, wui, Rj)
By using the same technique as in the previous section, we can calculate dv(i)/di foriin (1, i∗)
and (i∗,2). dv di =a sgs0 gs Ri gs(i)ws fori ∈(i∗,2) (13) dv di =a ug0u gu ( Ri guwu ) for i∈(1, i∗) (14)
Next we will check a single crossing property of the utility function U(p2, X)−Z(R, ws, wu, R).
The marginal rate of substitution betweenX and R is
MRS(R,x) = 1 Ux as gsws fori∈(i∗,2) = 1 Ux au guwu fori∈(1, i∗)
Thus the MRS(R,X) is a decreasing function of i and a single crossing property is satisfied.
This means that the local incentive compatibility and the monotone condition ofRare sufficient
conditions for the global incentive compatibility (Fudenberg and Tirole, 1991). We assume that
the monotonicity constraint is not binding.
As in the previous section, it is useful to think that the government controls v(i) andRi and
thatXi is defined from the following relationship:
v(i) =U(p2, X)−Z(wis, wiu, Rj)
Finally for analytical convenience, rewrite the first order condition of (13) and (14) :
˙ vs= g 0 s gs ashsi and v˙u = g 0 u gu auhui.
W(σ) = max Z i∗ 1 vu(i)nidi+ Z 2 i∗ vs(i)nidi st. v˙s= g 0 s gs ashsi fori∗ < i≤2 (IC1) ˙ vu = g 0 u gu auhui for 1< i < i∗ (IC2) vs(i∗) =vu(i∗) (BD1) Rsi∗=Rui∗ (BD2) Z 2 i∗ {Rsi −x(Rsi, vis, wis, wiu)}nidi + Z i∗ 1 {Rui −x(Rui, viu, wis, wiu)}nidi +σ{ Z 2 1 nic2idi−Y(p∗2+σ, Hs, Hu)} ≥0 (RC) where Hs= Z 2 i∗ hsigs(i)nidi andHu = Z i∗ 1 huigu(i)nidi
The above programming problem deserves several comments. First, (IC1) and (IC2) are the
local incentive compatibility constraints. Second, (BD1) comes from the assumption that the tax
schedule that the government designs is continuous and, as a result, the utility level of the agents
must be continuous. (BD2) comes from the assumption that individual i∗ chooses only one R.
Now let µsi,µui and λ be the Lagrangian multipliers of (IC1),(IC2) and (RC). Let β1 and β2 be
the Lagrangian multipliers of (BD1) and (BD2). The first order conditions can be calculated
and we will write them in the Appendix to save the space. Then, what we need to know is the
effect of increasing σ from zero on the social welfare, which is equivalent to dW/dσ. By using
the envelope theorem, we have (See Appendix)
dW dσ σ=0 = ∂i ∗ ∂σ µsi∗ashsi∗ gs0 gs −µui∗auhui∗ gu0 gu (15)
From the FOC of vis∗ and vui∗, we have µsi∗ = µui∗. In addition, as we show in the Appendix µsi
slope of vsi and the left hand slopeviu ati∗. From Lemma 1, the slope ofvis is steeper than the
slope ofviu ati∗. Since ∂i∂σ∗ >0, we havedW/dσ >0.
Proposition 2 Consider a small open economy where individuals accumulate human capital
endogenously and different types of human capital are perfect substitutes. Suppose that the social
planner designs a nonlinear income tax system to maximize the utilitarian social welfare function
without any production distortion and that there is no-bunching at the switching point i∗. Then,
introducing a tariff on an unskilled human capital intensive good will increase the social welfare.
At this point, it would be useful to consider the economic meaning of (15). Figure 1 shows
the graph of ev(i) , ves(i) and evu(i). When the government increases the tariff σ from zero, the
graph of evs(i) will shift downward and the graph of evu(i) shifts upward. As a result, i
∗ will
increase. Also, notice that from (IC1) and (IC2), the slope of evs(i) increases and the slope of
e
vu(i) decreases.
In the mechanism designs problem, ˙v, the slope of the value function, is related with how the
compensation schedule must be sensitive with unobserved ability. When ˙vis higher, it means that
the social planner needs to give higher utility to those with higher ability. With redistributive
social welfare function, the social planner wants to give higher utility to agents with lower ability.
Thus, when ˙vis high, the level of utility that the social planner can give to the agents with lower
ability is limited since the amount of the resource is limited. In such a situation, if the government
can make ˙v smaller exogenously, it is possible to increase the social welfare and changingσ can
be a good policy tool for changing ˙v.
When σ increases, the change of ˙v is not the same for all individuals however. As Figure 1
shows, all individuals whose ability is lower than i∗ will experience a decrease of ˙v and all
indi-viduals whose ability is greater thani∗ will experience an increase of ˙vexcept the neighborhood
of i∗. But, as the analysis in the Appendix shows, the effect of a change of ˙v for those agents is
the other hand, there are some individuals who experience the first order change of ˙v.
Individ-uals whose ability is in (i∗, i∗+∂i∗/∂σ) will switch from accumulating skilled human capital to
unskilled human capital. Since the graph ofv(i) has a counter-clockwise kink ati∗, individuals
in (i∗, i∗+∂i∗/∂σ) will experience the first order decrease of ˙v. This implies that the government
needs less ability-sensitive compensation schedules for those agents. Because this change of ˙v has
the first order effect, it will increase the social welfare.
(15) can be interpreted in terms of the marginal tax schedule as well. Note that (∂Z/∂Rm)/Ux
is equal to 1−TimwhereTimis the marginal tax rate of income of those who accumulatedm=s, u
type of skill and his ability is equal to i. From the FOC of Ris andRui,
λnTis=µsi ×as∂h s i ∂Rs i gs0 gs and λnTiu=µui ×au∂h u i ∂Ru i gu0 gu
Thus, Since (∂h/∂R)×R=h, we have
dW dσ σ=0 = ∂i ∗ ∂σλn(R s i∗Tis∗−Rui∗Tiu∗).
Tis∗ andTiu∗ are the marginal tax rates of individuals just above i∗ and just belowi∗, respectively.
Whenσincreases, the individual just abovei∗who initially accumulated skilled human capital will
switch from accumulating skilled human capital to unskilled human capital. Since the marginal
tax rate of those who accumulated skilled human capital is higher than the marginal tax rate for
those who accumulated unskilled human capital aroundi∗, the marginal tax rate will decrease.9 Thus,Rsi∗Tis∗−Rui∗Tiu∗ is the earning that is affected by a change of the marginal tax rates. Since
this change of the marginal tax rate is of the first order, it can increase the social welfare.
4
Discussion
In the above two sections, we have shown that indirect redistribution through an increase of the
return from unskilled human capital and a decrease of the return from skilled human capital
9
Readers still might wonder why the marginal tax rate for those who accumulated skilled human capital is higher than those who accumulated unskilled human capital aroundi∗. The reason is around the right hand side ofi∗, the marginal return from ability is higher at the right hand side ofi∗than at the left hand side ofi∗because
would increase the social welfare. One natural question at this point would be why such changes
of returns do not cause the adverse effect on human capital accumulation and, if they cause it,
why we can ignore it. The answer to such a question is that it will cause the adverse effect on
human capital accumulation but a redistributive income tax system also causes such an incentive
problem. In a circumstance where each individual’s comparative advantage is not observable
and human capital accumulation is endogenous, the redistributive income taxation necessarily
introduces adverse incentive effects on human capital accumulation. In such a situation, the
question is not whether redistributive change of returns from different types of skill causes the
adverse incentive effect but whether it can mitigate the existing incentive problem caused by
income taxation. Given that income taxation is subject to asymmetric information due to the
unobservability of comparative advantage at individual levels, redistributive changes of return
from different types of skill mitigates the asymmetric information problem since the government
can affect agents with different types of comparative advantage differently. As a result, the
redistributive changes of returns from different types of skill will increase the economic efficiency.
At this point we should emphasize that the assumption of comparative advantage plays a
crucial role in our analysis. This implies that empirical studies that examined the returns from
human capital accumulation for individual with different abilities such as Dinardo and Tobias
(2001) and Tobias (2003) are important. In addition, the results in the empirical studies and the
result in this paper can have important implications for public policy. For example, it might be
possible that encouraging skilled human capital accumulation through government funding does
not necessarily increase the social welfare if comparative advantage in human capital accumulation
exists and individuals who get the benefit most from the government funding are individuals with
higher ability.
In this paper, we examined the production efficiency theorem in a small open economy setting.
It is worth mentioning that in a small open economy setting, the Atkinson and Stiglitz theorem
(Samuelson, 1949). On the other, it is possible to extend the intuition of the present paper to a
closed economy setting by using the two sector-two factor general equilibrium model (Harberger,
1962). In this case, we can prove that the Atkinson and Stiglitz theorem does not hold since
a commodity tax can affect the producer prices and factor prices in a closed economy. More
specifically, we can prove that imposing a commodity tax on skilled human capital intensive good
will increase the social welfare.10
5
Conclusion
In this paper, we have examined whether indirect redistribution such as tariffs and production
subsidies can complement income taxation in the long run where human capital accumulation is
endogenous. For that purpose, I developed two models where individuals can choose the amount of
both skilled and unskilled human capital based on theircomparative advantage. In the first model,
we assumed that skilled human capital and unskilled human capital are imperfect substitutes and
that individuals accumulate both skilled and unskilled human capital. In the second model, we
assumed that skilled human capital and unskilled human capital are perfect substitutes and that
individuals accumulate only one type of human capital. Assuming that individuals with higher
ability have comparative advantage in accumulating skilled human capital, we have shown that
indirect redistribution such imposing a tariff on an unskilled human capital intensive good can
increase the efficiency and complement an income tax system. This suggests that the validity
of the production efficiency theorem depends on how the process of human capital accumulation
is modelled. The result of this paper also suggests that empirical studies such as Dinardo and
Tobias (2001) and Tobias (2003) that showed the returns from human capital were different
among individuals with different abilities have important implications for public policy.
10
Appendix
Derivation of equation (11)
Letµi andλbe the Lagrangian multiplier of the incentive constraint and the resource constraint.
Then, the Lagrangian function is
W(σ) = Z 2 1 v(i)nidi+ Z 2 1 µi[ dv di −f 0 s(hsi)hsi(g 0 s/gs)−fu0(hui)hui(g 0 u/gu)di]+ +λ Z 2 1 ni{Ri−xi(vi)}di+σ{ Z 2 1 nic2idi−y2(p∗2+σ, Hs, Hu)}
By using the integration by parts, we can obtain
W(σ) = Z 2 1 vinidi+ Z 2 1 µi dv didi− Z 2 1 µifs0(hsi)hsi(gs0/gs)di− Z 2 1 µifu0(hui)hui(gu0/gu)di +λ Z 2 1 ni{Ri−xi}di+σ Z 2 1 nic2idi−σY(p∗2+σ, Hs, Hu)} = Z 2 1 vinidi+µ2v2−µ1v1− Z 2 1 ˙ uividi− Z 2 1 µifs0(hsi)hsi(gs0/gs)di− Z 2 1 µifu0(hui)hui(g0u/gu)di +λ Z 2 1 ni{Ri−xi}di+λσ Z 2 1 nic2idi−Y(p∗2+σ, Hs, Hu)
Therefore, the first-order-conditions are
ni−u˙i−λni ∂xi ∂vi +λσ∂c2i ∂xi ∂xi ∂vi = 0 −µi d[fs0(hsi)hsi(g0s/gs)] dhsi ∂hui ∂Ri −µi d[fu0(hui)hui(gu0/gu)] dhui ∂hui ∂Ri +λni−λni ∂xi ∂Ri +λσni ∂c2i ∂xi ∂xi ∂Ri = 0 µ1 = 0 andµ2 = 0
By using the envelope theorem, we obtain
dW dσ σ=0 =− Z 2 1 µi d[fs0(hs i)hsi(g 0 s/gs)] dhsi { dhs i dwsi dws i dσ + dhs i dwui dwu i dσ }di − Z 2 1 µi d[fu0(hui)hui(g0u/gu)] dhs i {dh u i dws i dwsi dσ + dhui dwu i dwui dσ }di +λ{ Z 2 1 c2inidi−Y(p∗2+σ, Hs, Hu)}+λ Z 2 1 (−∂xi ∂p − ∂xi ∂wis ∂wis ∂σ − ∂xi ∂wu ∂wu ∂σ )nidi
Note that −∂xi/∂p2 = (Up2)/(Ux). From the Roy’s identity, (Up2)/(Ux) = −c2i. Therefore,
λR2 1 c2inidi=λ R2 1(− ∂xi ∂p)nidi. In addition, ∂xi ∂ws i =zw s i/Ux and ∂xi ∂wu =zwu/Uxand ∂xi ∂Ri =ZRi/Ux.
Using the definition of Zws i and Zwu, ∂xi ∂ws i = −αihsi/Ux, ∂w∂xiu =−αihui/Ux and ZR/Ux =αi/Ux .
On the other hand, the FOC of Ri atσ= 0 is that
−µi d[fs0(hsi)hsi(gs0/gs)] dhs i ∂hsi ∂Ri −µi d[fu0(hui)hui(gu0/gu)] dhu i ∂hui ∂Ri +λni =λniαi/Ux Thus,dW/dσ becomes dW dσ σ=0 =− Z 2 1 µi d[fs0(hsi)hsi(gs0/gs)] dhsi { dhsi dwsi dwis dσ + dhsi dwiu dwui dσ }di − Z 2 1 µi d[fu0(hui)hui(g0u/gu)] dhs i {dh u i dws i dwsi dσ + dhui dwu i dwui dσ }di −λy2+ Z 2 1 [−µi d[fs0(hsi)hsi(gs0/gs)] dhsi ∂hsi ∂Ri −µi d[fu0(hui)hui(g0u/gu)] dhui ∂hui ∂Ri +λni]hsi ∂wsi ∂σ di + Z 2 1 [−µi d[fs0(hsi)hsi(g0s/gs)] dhsi ∂hsi ∂Ri −µi d[fu0(hui)hui(gu0/gu)] dhui ∂hui ∂Ri +λni]ihui ∂wu ∂σ di −λy2+ Z 2 1 λnihsi ∂wsi ∂σ di+ Z 2 1 λnihui ∂wu ∂σ di Note thatR2 1 λnih s i ∂ws i ∂σdi+ R2 1 λnih u i ∂w u ∂σ di= R2 1 λnih s ii ∂ws ∂σ di+ R2 1 λnih u i∂w u ∂σ di. R2 1 λnih s ii∂w s ∂σ di+ R2 1 λnih u i∂w u
∂σ dis a change of total earning due to a tariff when levels of human capital of all
indi-viduals are fixed. On the other hand, from perfect competition, for given level of human capital
of all individuals, the total revenue of the firm should be equal to the total payment to
fac-tor owners. Thus, y1 + (p∗2 +σ)y2 = ws
R2 1 λnih s iidi+wu R2 1 λnih u
idi always holds. Let Q(σ)
be the total revenue of firms when all human capital level of all individuals are fixed. Then,
dQ/dσ=R12λnihsii∂w s ∂σ di+ R2 1 λnihui ∂w u ∂σ di. By definition of Q(σ) Q(σ) = max y1+ (p∗2+σ)y2 s.t. (y1, y2)∈Γ(Hs, Hu) = 0
Hs and Hu are fixed.
From the envelope theorem, dQdσ =y2. Therefore ,−λy2+
R2 1 λnihsi ∂wsi ∂σ di+ R2 1 λnihui ∂w u ∂σ di= 0.
Note that from the definition ofehsi and ehui, we have
fs0(ehsi) ∂ehsi ∂wsi +f 0 u(ehui) ∂ehui ∂wis = 0 andf 0 u(ehsi) ∂ehsi ∂wiu +f 0 u(ehui) ∂ehsi ∂wiu = 0
By using the Slutsky equation for hsiand hui and the above equation, we have dW dσ σ=0 =− Z 2 1 µi [f 00 s(hsi)hsi f0 s(hsi) + 1](gs0/gs)−[ fu00(hui)hui f0 u(hui) + 1](gu0/gu) fs0(hsi)∂eh s i ∂ws i ∂wis ∂σ di − Z 2 1 µi [f 00 s(hsi)hsi f0 s(hsi) + 1](gs0/gs)−[ fu00(hui)hui f0 u(hui) + 1](gu0/gu) fs0(hsi)∂eh s i ∂wui ∂wiu ∂σ di =− Z 2 1 µi [f 00 s(hsi)hsi f0 s(hsi) + 1](gs0/gs)−[ fu00(hui)hui f0 u(hui) + 1](gu0/gu) fs0(hsi)[∂eh s i ∂ws i ∂wsi ∂σ + ∂ehsi ∂wu i ∂wiu ∂σ ]di
From the condition of the comparative advantage, the inside of the large bracket is positive.
Also, both ∂ehsi ∂ws i ∂ws i ∂σ and ∂ehsi ∂wu i ∂wu i
∂σ are positive. Thus, we have dW dσ σ=0 >0 . Proof of µi ≥0
From the FOC ofvi, we will have ni−u˙i−λni∂x∂vii +λσ∂c∂x2ii∂x∂vii = 0. Thus, we have
ni−λni
∂xi
∂vi
= ˙µi
atσ= 0. By integrating both sides and using the definition of ∂xi
∂vi and µ1= 0, we will have
Z i 1 ni{1− λ Ux }=µi
From the first order condition of the revelation problem, Ux(p2, X)X0(i) =ZRR0(i). This means
that the sign of X0(i) and R0(i) are the same. Since v(i) is strictly increasing, X0(i) and R0(i)
must be increasing. When X0(i) is increasing, Uλ
x is increasing. This implies that if at some i
∗∗,
1−λ/Ux = 0, then for any i > i∗∗, 1−λ/Ux <0. However, µ2 = 0 from the FOC of v2. This
implies that µ1 is initially strictly positive untili∗∗ and then it begins to decrease and reaches
to zero at i= 2. Therefore,µi≥0 for all 1≤i≤2.
The Lagrangian is: L= Z i∗ 1 vu(i)nidi+ Z 2 i∗ vs(i)nidi+ Z i∗ 1 µui{v˙u−auhu i(g0u/gu)}di+ Z 2 i∗ µsi{v˙s−ashs i(gs0/gs)}di +β1{vis∗−vui∗}+β2{Rsi∗−Rui∗} +λ Z i∗ 1 {Rui −x(Rui, vui, wis, wiu)}nidi+λ Z 2 i∗ {Rsi −x(Rsi, vis, wsi, wui)}nidi +λσ{ Z 2 1 nic2idi−Y(p∗2+σ, Hs, Hu)}
By using the integration by parts, we obtain
L= Z i∗ 1 vu(i)nidi+ Z 2 i∗ vs(i)nidi+µui∗viu∗−µu1v1u− Z i∗ 1 ˙ µuivuidi− Z i∗ 1 µuiauhui(gu0/gu)di µs2v2s−µsi∗vsi∗− Z 2 i∗ ˙ µsivsidi− Z 2 i∗ µisashsi(gs0/gs)di+β1{vis∗−viu∗}+β2{Rsi∗−Rui∗} +λ Z i∗ 1 {Rui −x(Rui, viu, q, wsi, wui)}nidi+λ Z 2 i∗ {Ris−x(Rsi, vis, wis, wui)}nidi +λσ{ Z 2 1 nic2idi−Y(p∗2+σ, Hs, Hu)}
The first order condition forvis,v2s,vis∗,Rsi,Ris∗,viu,vui∗,v1u,Rui and Rui∗ are
vis:ni−µ˙si −λni ∂x ∂vis +λσ ∂c2i ∂xi ∂xi ∂vis = 0 vs2 :µs2 = 0 vsi∗ :−µsi∗+β1 = 0 Rsi :−µsi ×as∂h s i ∂Rs i gs0 gs +λni−λni ∂x ∂Rs i +λσni ∂c2i ∂xi ∂xi ∂Ri = 0 Rsi∗ :β2= 0 vui :ni−µ˙si −λni ∂x ∂vis +λσ ∂c2i ∂xi ∂xi ∂vsi = 0 viu∗:µui∗−β1 = 0 vu1 :µu1 = 0 Rui :−µui ×au∂h u i ∂Ru i gu0 gu +λni−λni ∂x ∂Ru i +λσni ∂c2i ∂xi ∂xi ∂Ru i = 0 Rui∗ :β2= 0
Now we characterize those first order conditions. First, note that atσ = 0, µsi =µsi∗+ Z i i∗ nj(1−λ ∂x ∂vj )dj fori∈(i∗,2) and µui∗ =µu1 + Z i∗ 1 nj(1−λ ∂x ∂vj )dj (16) Thus, since µu 1 = 0, µsi = Ri 1nj(1−λ ∂x ∂vj)dj fori ∈(i ∗,2). Note that ∂x ∂vj = 1/(Ux) . A single crossing property and the monotonicity ofRsi andRui guarantee thatxiis increasing. This implies
that ∂v∂x
j is increasing and the inside of the integral is a decreasing function of i. Since µ
s
2 = 0,
µui and µsi are non-negative.
Now we examine dW/dσ and evaluate at σ = 0.From the envelope theorem,
dW dσ = ∂i∗ ∂σ vu(i∗)ni∗−vs(i∗)ni∗+ ˙µui∗vu(i∗) +µui∗v˙iu∗−µ˙ui∗vu(i∗)−µui∗auhui gu0 gu −µsi∗v(i∗)−µsi∗v˙si∗+ ˙µsi∗v(i∗) +µsi∗ashsi gs0 gs +β1{v˙si∗−v˙u i∗}+β5R˙ui∗−β5R˙si∗ −λ{Rsi∗−x(Rsi∗, vis∗, q, wis∗, wui∗)}ni∗+λ{Ru i∗−x(Rui∗, vui∗, q, wis∗, wui∗)}ni∗ +∂w s ∂σ ( − Z 2 i∗ µsias∂h s i ∂ws i g0sdi−λ Z 2 i∗ ∂x ∂ws i nigs(i)di ) +∂w u ∂σ ( − Z 2 i∗ µuiau∂h u i ∂wu i gu0di−λ Z i∗ 1 ∂x ∂wugu(i)nidi ) +λ Z 2 1 nic2idi−λY(p∗2+σ, Hs, Hu) + Z 2 i∗ −∂x ∂q ∂q ∂σnidi− ∂x ∂q ∂q ∂σ Z i∗ 1 nidi
From the Roy’s identity, c2i =−∂x∂q∂q∂σ. Thus,
dW dσ = ∂i∗ ∂σ µsi∗ashsi g0s gs −µui∗auhui gu0 gu +∂w s ∂σ ( − Z 2 i∗ µsias∂h s i ∂wsig 0 sdi−λ Z 2 i∗ ∂x ∂wsnigs(i)di ) +∂w u ∂σ ( − Z 2 i∗ µuiau∂h u i ∂wiug 0 udi−λ Z i∗ 1 ∂x ∂wunigudi ) −λY(p∗2+σ, Hs, Hu)
Now we need to calculate the inside of the integral. Note that from the definition of hsi and hui,
we have ∂hsi ∂wsi =−h s i ∂hsi ∂Rsi and ∂hui ∂wiu =−h u i ∂hui ∂Riu This implies that
−µsias∂h s i ∂wisg 0 s=µsiashsi ∂hsi ∂Rsig 0 s and −µuiau ∂hui ∂wui g 0 u =µuiauhui ∂hui ∂Rui g 0 u
By using the FOC of Rsi and Riu, µsiashsi ∂h s i ∂Rs i gs0 =hsigs{λni−λni ∂x ∂Rs i } µuiauhui ∂h u i ∂Rui g 0 u =huigu{λni−λni ∂x ∂Rui } Thus, dWdσ is dW dσ = ∂i∗ ∂σ µsi∗ashsi gs0 gs −µui∗auhui gu0 gu +∂w s ∂σ Z 2 i∗ hsigs{λni−λni ∂x ∂Rs i }di−λ Z 2 i∗ ∂x ∂ws i gsnidi +∂w u ∂σ ( Z 2 i∗ huigu{λni−λni ∂x ∂Rui }di−λ Z i∗ 1 ∂x ∂wui gunidi ) −λY(p∗2+σ, Hs, Hu)
Next, we need to calculate λ∂w∂σsR2
i∗hsigs(i)ni+λ∂w
u
∂σ Ri∗
1 huigu(i)nidi. From the argument of the
previous section, we have
λy2 =λ ∂ws ∂σ Z 2 i∗ hsigs(i)ni+λ ∂wu ∂σ Z i∗ 1 huigu(i)nidi.
Third, we will show that hsi∂R∂xs i = −∂w∂xs i and h u i ∂R∂xu i = −∂w∂xu
i. From the definition of Z, we have ∂Z ∂Rs i =as/wsi and ∂Z ∂ws i =−ashsi(1/wis) for i∈(i∗,2) ∂Z ∂Rui =a u/wu iand ∂Z ∂wui =−a uhu i(1/wis) for i∈(1, i ∗ )
Thus, by using the definition of ∂R∂xs i,
∂x ∂ws ,∂R∂xs
i,
∂x
∂ws, we can check thathsi∂R∂xs i = −∂w∂xs i andh u i ∂R∂xui = −∂w∂xu i . Therefore, dW/dσ is dW dσ σ=0 = ∂i ∗ ∂σ µsi∗ashsi∗ gs0 gs −µui∗auhui∗ gu0 gu
From the FOC of vis∗ and viu∗, we haveµis∗ =µui∗. In addition, ashsig 0 s gs and a uhu i g0 u
gu are the right side slope ofvisand the left side slope vui ati∗ From Lemma 1, the slope ofvis is steeper than the
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